Lower-Dimensional Manifold (Algebraic) Representation of Reynolds Stress Closure Equations |
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Authors: | Sharath S Girimaji |
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Institution: | (1) Aerospace Engineering Department, Texas A & M University, College Station, TX 77843-3141, USA, US |
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Abstract: | For complex turbulent flows, Reynolds stress closure modeling (RSCM) is the lowest level at which models can be developed
with some fidelity to the governing Navier–Stokes equations. Citing computational burden, researchers have long sought to
reduce the seven-equation RSCM to the so-called algebraic Reynolds stress model which involves solving only two evolution
equations for turbulent kinetic energy and dissipation. In the past, reduction has been accomplished successfully in the weak-equilibrium
limit of turbulence. In non-equilibrium turbulence, attempts at reduction have lacked mathematical rigor and have been based
on ad hoc hypotheses resulting in less than adequate models.?In this work we undertake a formal (numerical) examination of the dynamical
system of equations that constitute the Reynolds stress closure model to investigate the following questions. (i) When does
the RSCM equation system formally permit reduced representation? (ii) What is the dimensionality (number of independent variables)
of the permitted reduced system? (iii) How can one derive the reduced system (algebraic Reynolds stress model) from the full
RSCM equations? Our analysis reveals that a lower-dimensional representation of the RSCM equations is possible not only in
the equilibrium limit, but also in the slow-manifold stage of non-equilibrium turbulence. The degree of reduction depends
on the type of mean-flow deformation and state of turbulence. We further develop two novel methods for deriving algebraic
Reynolds stress models from RSCM equations in non-equilibrium turbulence. The present work is expected to play an important
role in bringing much of the sophistication of the RSCM into the realm of two-equation algebraic Reynolds stress models. Another
objective of this work is to place the other algebraic stress modeling efforts in the lower-dimensional modeling context.
Received 19 November 1999 and accepted 3 August 2000 |
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